I have a long expression that has absolute values in the exponent. Some parts of the expression would clearly simplify if one assumes that the power is an odd or even integer. When I give one of this assumption, Mathematica has a very puzzling behaviour. I will give four examples and I would be grateful if someone could explain

Simplify[1+(-1)^Abs[j],{(j+1)/2 ∈ Integers}]

Simplify[t^Abs[j](t+(-1)^Abs[j]),{(j+1)/2 ∈ Integers}]

Simplify[t^Abs[j](1+(-1)^Abs[j]),{(j+1)/2 ∈ Integers}]

Simplify[t^Abs[j](1+t+(-1)^Abs[j]),{(j+1)/2 ∈ Integers}]

The first expression Mathematica rightly simplifies to 0.

For the second expression Mathematica doesn't simplify what is in the brackets, even though it can - maybe Mathematica thinks that if it cannot simplify one absolute value, then it shouldn't simplify any?

The third expression however simplifies, maybe because it manages to simplify to 0?

The fourth expression is the most puzzling. Here Mathematica refuses to simplify what is in the brackets. Presumably it doesn't see it as a simplification? Why?

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